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Theorem hlbn 19186
Description: Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Assertion
Ref Expression
hlbn  |-  ( W  e.  CHil  ->  W  e. Ban )

Proof of Theorem hlbn
StepHypRef Expression
1 ishl 19185 . 2  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
21simplbi 447 1  |-  ( W  e.  CHil  ->  W  e. Ban )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   CPreHilccph 19002  Bancbn 19157   CHilchl 19158
This theorem is referenced by:  hlcms  19189  hlprlem  19190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-in 3272  df-hl 19161
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